Noether symmetries of F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} cosmology

Abstract

This article explores an analysis via the Noether symmetry approach to investigate an extended teleparallel F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} gravity model in the context of a Friedmann–Robertson–Walker spacetime. The model incorporates the torsion scalar T, scalar field ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} and the kinetic term X of the scalar field. This approach allows us to select physically interesting models by restricting arbitrariness in the F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} function. Thus we focus on two types of the function F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document}, namely models involving the generalized teleparallel dark energy and the nonlinear function of the kinetic term and analyze the Noether symmetry properties and demonstrate the presence of non-vanishing Noether vectors. In the first model, by specifically considering the X4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf{X_{4}}$$\\end{document} Noether symmetry, representing a scaling symmetry, we simplify the field equations and introduce a cyclic variable to enable a more workable formulation. Furthermore, we present a comprehensive graphical analysis, showcasing the time evolution of the scale factor and the scalar field by using the solutions obtained through these new variables. Additionally, we examine some fundamental cosmological parameters. For the second model, which includes the nonlinear form of the kinetic term, Noether symmetry results in obtaining simple power-law solutions for the dynamic variables. Our findings clearly show that the considered models represent an accelerating expansion of the Universe, which is consistent with observations. This study highlights the efficacy of the Noether symmetry approach in defining the functional form of F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} and obtaining precise cosmological solutions. The analytical solutions and graphical analysis offer valuable insights into the dynamic behavior of the models, contributing to understanding the evolving nature of the Universe.